Modulation method and apparatus for static power frequency changers

ABSTRACT

The performance of the cycloconverter when used to drive an induction motor is at present limited because of inadequate methods of modulating the thyristor conduction angles. The main problems are: low maximum frequency due to subharmonics; voltage distortion due to discontinuous current and to uncertainty of current cross-over (for non-circulating current mode); and poor power factor on the input. This specification describes a modulation method which overcomes the first two of these problems without sacrificing performance in any area and proposes a new method to greatly improve the input power factor, although at the expense of possible input current subharmonics. A non-limiting application of the method to a 3 pulse, 3 phase, non-circulating current cycloconverter is described. The technique may also be applied to static power frequency changers other than the cycloconverter. In one broad form, there is provided in a static power frequency changer connecting one or more input phases to one or more outputs, said changer comprising one or more electronic switching means comprising a plurality of electronic switches, modulation means to sequentially activate individual switches of said electronic switching means, said electronic switching means connecting an AC voltage supply comprising one or more input phases to an output, so that the output voltage waveform at said output is built up of sections of the input voltage waveforms on said one or more input phases; a method of selecting, for each said output, an instant of switching of the input waveform to be connected to said output, wherein: for each output said instant of switching is chosen so that during a predetermined time interval the average of the difference between the continuous integral of the desired output voltage and an estimate of the continuous integral of the actual output voltage is minimized, said predetermined time interval including said instant of switching to another input waveform.

BACKGROUND OF THE INVENTION

The present invention relates principally to cycloconverters but also toother static power frequency changers, and, in particular, to specificmethods of and apparatus for switching input waveforms from an AC supplyof one or more input phases to achieve approximations to desired outputwaveforms.

NOMENCLATURE

f_(i) Input frequency.

f_(o) Output frequency.

t₁ The starting time of a trigger period.

t₂ The ending time of a trigger period.

t₀ The time of occurrence of a particular phase angle of V_(t) for agiven trigger period.

t_(f) The time the thyristor is triggered in a trigger period.

t_(c) The time the current drops to zero (if this occurs) in a triggerperiod.

v_(o) Output voltage on one phase of the cycloconverter.

v_(r) Output reference voltage on one phase.

v_(b) Boost voltage in one phase applied to overcome IZ voltage drop.

v_(t) The input voltage connected to the thyristor to be triggered in atrigger period.

v_(p) The input voltage connected to the thyristor that is on at thestart of a trigger period.

ψ Reference value of flux linkage in one phase of the induction motor.

K A constant determining stability.

L₁ Per phase stator leakage inductance.

L₂ Per phase, stator referred, rotor leakage inductance.

PRIOR ART

A static frequency changer is essentially a device for synthesizing anapproximation to a desired output waveform by means of switching oneportion of one or more input waveforms consecutively to the output ofthe device. The input waveforms used are either the input phase voltagewaveforms or the inversion of these or both. The number of inputwaveforms used is called the pulse number of the frequency changer, socalled because this is usually (although not necessarily) the averagenumber of portions of the input waveforms switched to the output overone input cycle. The desired output waveform will typically have afrequency of less than half that of any input waveform. Switching of theinput waveform is typically done at the input waveform frequency.

Cycloconverters can be defined as static frequency changers which usethyristors that are naturally commutated. Cycloconverters may be eitherof the circulating current type or the non-circulating current type. Allother types of static frequency changers presently use switches thateither have the intrinsic ability to turn off or use thyristors whichare turned off with forced commutation.

Known input waveform switching strategies include "cosine crossingcontrol" and "integral control".

Cosine crossing control uses a switching criteria based upon theintersection of selected portions of a phase shifted input waveform(typically 90°) and the desired output reference waveform. The integralmethod is based on the selection of input waveform triggering instantsdetermined when the integral of the difference between the outputvoltage waveform and the desired reference voltage waveform (determinedin real time) is equal to zero. The limitations of both these methods asapplied to the cycloconverter are discussed in U.S. Pat. No. 3,585,485to Gyugyi, Rosa and Pelly. In U.S. Pat. No. 3,585,485 a particularsolution to an inherent problem in applying the integral method toapproximate a non DC output waveform is disclosed. The solution involvesinjecting an offset component into the next integral calculation, theoffset component being proportional to the DC component of thecalculated ripple integrals of the integral method. U.S. Pat. No.3,585,485 is concerned with the application of the integral method to acirculating current type cycloconverter. U.S. Pat. No. 3,585,486 is aconcurrent patent to the same inventors concerned with applying theintegral method to the non-circulating current cycloconverter.

PROBLEMS SOUGHT TO BE OVERCOME AND ADVANTAGES OF PREFERRED EMBODIMENTS

The method of the present invention has particular applicability to butis not solely limited to non-circulating current cycloconverters.

The non-circulating current cycloconverter has many advantages overother forms of A.C. variable speed drives: its maximum power output isvirtually unlimited; its power circuit is very simple, consisting ofonly phase-controlled thyristors and their associated snubbers; it isvery efficient; and it is naturally regenerative. With the presentmodulation methods in use, however, (cosine crossing control/integralcontrol-with or without feedback) it suffers from some severedisadvantages. It has a low maximum output frequency (of about 25 Hz fora 6-pulse system) due to subharmonics appearing on the output. Itsuffers from voltage distortion and the associated torque pulsations dueto the uncertainty of the current cross-over points and the inability ofthe prior art modulation methods to compensate for discontinuouscurrents in the thyristors. Also, it has a poor input power factor,particularly at low output voltages.

The performance can be improved by adding an extra current feedback looparound the cycloconverter and its modulator (refer for example to HAkagi et al "Application of microcomputer to current controlledcycloconverter system: in Electrical Engineering in Japan Vol. 100, No.4, 1980, PP86-94). Using this approach, the improvement is limited bystability considerations, and the cycloconverter then becomes a currentcontrolled device, rather than the more ideal voltage controlled device.Current control is particularly a problem with multi-motor drives.

It is postulated herein and is considered advantageous to solve theperformance problems by improving the basic modulation method, ratherthan by attempting to linearize the present methods with currentfeedback.

The proposed modulation method of the present invention is an attempt toimprove the basic modulation method. With the proposed improved method,subharmonics are virtually, if not entirely, eliminated; the occurrenceof discontinuous current actually reduces the output voltage distortion,rather than increasing it; and cross-over between thyristor banks alwaysoccurs at the optimum time. The maximum output frequency using the newmethod is at least 25 Hz for a 3-pulse cycloconverter (refer Section3.4) and is expected to be 50 Hz for a 6-pulse cycloconverter.

The proposed improved modulation method is generally referred to hereinas double integration control.

Double integration control is particularly attractive when used with a3-pulse cycloconverter in an induction motor drive. The power circuitconsists of only 18 thyristors (refer FIG. 2) and has the sameefficiency and size as the equivalent converter for a D.C. motor drive.In Example 1 of a preferred embodiment, the performance is at least asgood as the equivalent 12 thyristor, 4-quadrant D.C. motor drive, withthe advantage of using the more rugged induction motor.

The price paid for the improved performance of the improved modulationmethod is a more complex control circuit. The double integration controlmethod is considerably more complex than the present methods based oncosine-wave crossing control. Microprocessor or equivalent dedicatedimplementation is considered essential. Example 1 of a preferredembodiment herein disclosed uses the extremely fast (200 ns instructiontime) 16 bit TMS32010 microprocessor from Texas Instruments. A slowermicroprocessor could be used, but at the sacrifice of response time (atpresent 7 ms for the 3-pulse system) and with an increase in currentripple.

At present, fully regenerative A.C. drives use either a P.W.M. inverterwith a fully controlled four quadrant bridge on the input or thesimpler, but lower performance, current source inverter. Thecycloconverter, with its high efficiency, simple power circuit, and highperformance has the potential if suitably driven to become the firstchoice in this application and in most four quadrant d.c. driveapplications. Disclosed herein is a method and apparatus for overcomingpresent problems with cycloconverters, particularly (although notexclusively) the 18 thyristor, 3-pulse non-circulating currentcycloconverter, that allow this potential to be achieved. However, theinvention is not to be construed as limited solely to such applications.The method is useful with other forms of static power frequency changerswhenever an improved voltage waveform leading to an improved currentwaveform is desired usually dictated by the nature of the load.

Power Factor Improvement

The improved modulation method does not improve the input power factor,but described herein is a way of improving the input power factor whichworks well within the modulation method of the present invention. Thepower factor improvement method, however, may cause subharmoniccomponents of the input current to appear, particularly in the case of athree pulse cycloconverter. The power factor improvement methoddisclosed herein can be utilized with any modulation scheme.

Pre-Integration Control

An alternative switching criteria which attempts to overcome theproblems of the two previously discussed methods (cosine crossing andintegral) is termed herein "pre-integration" control. Pre-integrationcontrol involves a selection of switching instants on the basis of theequality of calculated areas enclosed between the desired and actualoutput waveforms. Pre-integration control differs from integral controlin that part of the area required for determination of switchinginstants is not available in real time as it is in advance of theswitching instant and must be pre-calculated on the basis of an estimateof output waveform behaviour. Use of pre-calculation introduces inherentstability to this modulation method.

The pre-integration control method has advantages over the prior art.Its performance characteristics include:

1. It is stable (because of pre-calculation) as compared to the integralcontrol method:

2. Relative to the cosine control method:

(a) Pre-integration control compensates for discontinuous current,

(b) it virtually if not entirely eliminates sub-harmonic components ofthe output voltage (where sub-harmonic components are defined asfrequency components less than the desired output frequency),

(c) an induction motor can be unstable when controlled by the cosinecrossing method because discontinuous current is not compensated for.

3. Concerning bank cross over techniques the pre-integral method ensuresminimum voltage distortion during cross over between banks.

4. The cosine crossing control method with feed back is only partiallyeffective in reducing sub-harmonics. It is certainly much less effectivethan the pre-integral control method in this respect. Furthermore, withcosine crossing control with feedback, at high frequencies the feedbackhas to be reduced so that the method effectively reverts to ordinarycosine crossing control with its inherent sub-harmonic problems at highfrequency.

Problems with Pre-Integration Control Method

FIG. 7(a) shows the output waveform, v_(o), that would be obtained fromthe reference waveform, v_(r), using pre-integration control. It isassumed for simplicity that the output current is in phase with theoutput voltage and does not become discontinuous. FIG. 7(b) shows theintegral of v_(r) and the integral of v_(o). It can be seen that theaverage of the integral of the output waveform is badly distorted withthe pre-integration control method. In an induction motor, this wouldcause a corresponding distortion in the flux waveform which woulddegrade the performance of the motor.

Double Integration Control

The double integral modulation method of the present invention retainsthe advantages of pre-integration control, viz elimination ofsubharmonics and compensation for discontinuous current.

Bank Switching

For the naturally commutated cycloconverter the method of switchingbetween banks is also important in order to obtain satisfactory outputstherefrom.

To complement the improve modulation method of the present invention,disclosed herein are improved methods of determining optimum bankcrossover time.

Prior art methods have heretofore produced undesirable voltagedistortion of the output waveform due to poor selection of the bankcrossover time.

The most commonly applied prior art method adopts a bank crossoverselection criterion based on crossover at the instant when the outputcurrent first goes to zero during bank operation.

BRIEF DESCRIPTION OF THE INVENTION

The basis of the double integration method of the present inventionderives from the expression: ##EQU1## as applied to the modulation ofstatic power frequency changers.

In one broad form, the static power frequency changer of the inventionincludes a plurality of electronic switches connecting an AC voltagesupply having one or more input phases to one or more outputs. Amodulator sequentially activates individual electronic switches so thatthe output voltage waveform at each output is built up of sections ofthe input voltage waveforms on one or more of the input phases. A methodof selecting, for each output, an instant of switching of the inputwaveform to be connected to that output is provided, wherein for eachoutput the instant of switching is chosen so that during a predeterminedtime interval the average of the difference between the continuousintegral of the desired output voltage and an estimate of the continuousintegral of actual output voltage is minimized. The predetermined timeinterval includes the instant of switching to another input waveform.

In a further broad form of the invention, a method of bank switching ina cycloconverter is provided. The cycloconverter includes a plurality ofnaturally commutated thyristor divided into positive and negative banks,wherein only one bank is operating at any one time. The method ofchoosing the time of switching the thyristors includes a method ofadjusting the time of switching of the thyristors to take account ofdiscontinuous current. The method of bank switching comprises changingto an alternate bank when the time of switching is delayed to or past awaveform intersection time. The wave intersection time is defined as thetime of intersection of the desired output waveform with the inputwaveform to be switched to the output at the time of switching.

In yet a broader form, the invention provides a method of bank switchingin a cycloconverter having naturally commutated thyristors divided intopositive and negative current switching banks, wherein only one bank isoperating at any one time. The method includes selecting the instant ofcrossover from the positive bank to the negative bank so that it occursat the first instant that the output current is zero and, at the sametime, the integral of the difference between the output voltage and areference voltage is positive. The method further includes selecting theinstant of switching from the negative bank to the positive bank at thefirst instant when the output current is zero and, at the same time, theintegral of the difference between the output voltage and the referencevoltage is negative.

BRIEF DESCRIPTION OF DRAWINGS

Embodiments of the present invention will now be described withreference to the drawings in which:

FIG. 1 is a schematic diagram of the power circuit of a single inputphase, two pulse, two output phase cycloconverter, which uses triacssuitable to be driven by a preferred embodiment of the presentinvention, (the circuit being connected to a two phase, split windinginduction motor),

FIG. 2 is a schematic diagram of the power circuit of a three inputphase, three pulse, three output phase cycloconverter, suitable to bedriven by a preferred embodiment of the present invention,

FIG. 3 is a schematic diagram of the power circuit of a three inputphase, six pulse, three output phase cycloconverter, suitable to bedriven by a preferred embodiment of the present invention,

FIG. 4 graphically depicts one trigger period of the output of acycloconverter using pre-integration control,

FIG. 5 graphically depicts a practical method of implementingpre-integration control,

FIG. 6 graphically depicts the effect of discontinuous current on theoutput of a cycloconverter using pre-integration control,

FIGS. 7A and 7B graphically depict typical waveforms withpre-integration control (on the assumption that output current issinusoidal and in phase with the output voltage). FIG. 7A depicts outputand reference voltages (v_(o) and v_(r) respectively) shown with inputvoltages. FIG. 7B depicts the integrals of V_(o) and V_(r) from FIG. 7Ashowing the distortion produced with pre-integration control.

FIGS. 8A and 8B graphically provide an illustration of instabilityarising from unstabilised double integration control. FIG. 8A depictsoutput and reference waveforms v_(o) and v_(r) respectively with inputwaveforms. FIG. 8B depicts the integral of v_(o) and v_(r) (for the casewhere the integral of v_(r) is equal to 0).

FIG. 9A depicts a simple per phase equivalent circuit for ripple currentdetermination.

FIG. 9B depicts a typical waveform for the integral of the differencebetween the v_(o) and v_(r). (This corresponds to the ripple currentwaveform).

FIG. 10 graphically depicts optimum bank cross over time,

FIG. 11 is a power circuit schematic of a basic three pulsecycloconverter with motor load,

FIGS. 12A and 12B graphically depict the derivation of new outputvoltage reference waveforms by adjustment of neutral voltage reference.

FIG. 12A depicts output voltage reference waveforms and neutral voltagewaveform V_(n) with output neutral as reference. V_(n) is chosen to be1/2V_(a).

FIG. 12B depicts output voltage reference waveforms with V_(n) added.

FIG. 13 depicts a power circuit suitable to create an input reference(used in example 1),

FIGS. 14A and 14B, disclose typical output voltage ripple waveforms atmaximum positive voltage level for two possible choices of inputreference, (both diagrams are to the same scale). FIG. 14A shows ripplewaveforms using a neutral reference. FIG. 14B depicts ripple waveformsusing the reference of FIG. 13.

FIG. 15 is a block diagram of an example (Example 1) of a three inputphase, three pulse, three output phase cycloconverter utilisingpreferred embodiments of the method(s) of the present invention,

FIG. 16 graphically depicts an approximation waveform used forcalculations relating to stability.

FIGS. 17A and 17B depict changes to output reference waveforms for powerfactor improvement. FIG. 17A depicts output reference waveforms withzero nuetral voltage. FIG. 17B depicts the modified output referencewaveforms.

FIG. 18A depicts input waveforms showing the improved reference forpower factor correction.

FIG. 18B depicts input waveforms drawn with respect to the improvedreference of FIG. 18A with one phase of the output reference waveformsalso shown.

FIGS. 19A-D graphically depict typical waveforms for a three pulsecycloconverter using the power factor improvement method disclosedherein with an output amplitude 20% of maximum and an output frequency50% of the input frequency.

FIGS. 19A, B and C depict the voltage on each of the 3 output phasestogether with the reference voltage and the 3 input voltages.

FIG. 19D depicts the input current on the R phase together with the Rphase input voltage and the 3 output currents. The input currentwaveform was derived graphically from the other waveforms.

FIG. 20 depicts typical waveforms of the integrals of v_(o) and v_(r)for three different practical methods of implementing double integrationcontrol.

1. DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

For the purpose of the description, it is assumed that the positivecurrent thyristor bank is operating. For the negative current thyristorbank, the method is identical except the direction of the voltages andcurrents is reversed.

1.1 "Pre-integration control"

Described here initially is a modulation method termed pre-integrationcontrol. This modulation method can be used to advantage in A.C. to D.C.thyristor converter control.

The pre-integration control method is illustrated in FIG. 4. Shown arethe input voltage waveforms, the wanted fundamental output voltage,v_(r), and the trigger instant, t_(f), of the thyristor. In this controlscheme, the thyristor should be triggered between the instances when thereference waveform intersects the input waveforms fed to the incomingand outgoing thyristors. These two instances are the ideal start and endtimes of the period, called the trigger period, over which thecalculations for pre-integration control are carried out, although otherchoices of the trigger period can be used. In FIG. 4, the trigger periodchosen is the time from t₁ to t₂. The time t_(f) when the thyristor istriggered occurs when area C is equal to area D. Here, area D cannot bemeasured directly because it occurs after the thyristor is triggered. Itmust be pre-calculated. Expressed mathematically, t_(f) is chosen sothat: ##EQU2##

This scheme aims to keep the integral of an estimate of the outputvoltage from t₁ to t₂ equal to the integral of the reference voltageover the same time and thus attempts to keep the average of the twowaveforms over this interval the same. An advantage of this method fordriving an induction motor is that if the motor flux is at the correctvalue at time t₁, then it will also be at the correct value at time t₂and at the end of every subsequent trigger period. Also, because theintegrals of the reference and output waveforms are equal to each otherat the end of each trigger period, there can be no long term build up inerrors causing subharmonics. (Error build-up is a particular problemwith the prior art cosine-crossing method).

1.2 Compensating for Discontinuous Current--A Practical Implementationof Pre-Integration Control

The negative voltage excursions below v_(r) that occur before thetrigger instant, t_(f), when the output phase current is positive canbring this current to zero for a short time (i.e. make the currentdiscontinuous). A similar situation can occur for a negative phasecurrent due to positive voltage excursions above v_(r). During this timeof zero current, the phase voltage, v_(o), on the motor terminal dependson the voltages of the other two phases and is largely indeterminate.

An improved calculation method which results in the same triggering timeas the above method, but is more practical when the voltage distortioncaused by discontinuous current is to be compensated for is illustratedin FIG. 5. Area A in this figure is first pre-calculated. This is givenby the formula: ##EQU3##

Next, the voltage difference v_(t) -v_(o) is integrated in real timefrom time t₁. When this integral reaches the value of area A (which waspre-calculated), the thyristor is triggered. The algorithm is really theexpansion of equation (2) to the following: ##EQU4##

The first term is pre-calculated at the start of the trigger period. Theother term is calculated repeatedly from the start of the period witht_(f) replaced by the current time, t. When the second term reaches thevalue of the first term, the thyristor is triggered.

The effect of this method on discontinuous current is shown in FIG. 6.At time t_(c), the current drops to zero and the thyristor that waspreviously on turns off. With no connection between the input and outputof the cycloconverter, v_(o) is now indeterminate. The current remainszero until time t_(f) when the next thyristor triggers, re-establishinga positive current flow. As can be seen from FIG. 6, if the thyristor istriggered by the previously described method, i.e. when area B is equalto area A, the integral from t₁ to t₂ of v_(o) -v_(r) is again zero andthus the effects of the discontinuous current are compensated for.

1.3 Description of Double Integration Control Method

It can be seen from FIG. 7 that with pre-integration control the averageof v_(o) is maintained at the average of v_(r) over one trigger period,but this is not the case for the integrals of these two waveforms. Tomaintain correct flux in an induction motor, keeping the average of thetwo integrals equal is the main requirement. Expressed mathematically,this is achieved when t_(f) in a trigger period is chosen so that:##EQU5##

This is the basis of double pre-integration control.

The value of t_(f) which solves this equation depends on the initialvalue of the integral of (v_(o) -v_(r)) at the start of the period. Thisleads to problems in implementation in a practical situation (refer alsoto U.S. Pat. No. 3,585,485 where similar problems were encountered withthe integral modulation method). Unsymmetrical triggering, as shown inFIG. 8 can develop. This can be thought of as a form of systeminstability. The instability shows itself as an oscillation in the valueof the integral of v_(o) -v_(r) at the end of each trigger period.

To stabilise this modulation method, one technique is to make use of theobservation from FIG. 8(b) that the difference between the integral ofV_(o) -V_(r) at the end of consecutive periods oscillates when thesystem is unstable. Other techniques can be used. When the system isunstable, the trigger time of the thyristor can be adjusted to reducethis difference, and thus reduce the instability, at the expense ofletting the value of the RHS of equation 5 change from its ideal valueof zero. A practical way of achieving this is to change equation 5 byadding a suitable proportion of this difference to it so that the valueof the expression in equation 5 is forced to change in such a way as tosuppress the instability. Equation 5 with this incorporated becomes:##EQU6##

This is the technique used in example 1.

Note that the constant, K, has been multiplied by the term (t₂ -t₁) tomake it dimensionless. To determine the optimum value of the constant, arough computer simulation of the modulation method, which assumes theinput waveforms are trapezoidal rather than sinusoidal (this is closerto the actual waveforms in example 1) was carried out [Appendix I]. Itwas found that an optimum value of K, which corresponds to criticaldamping, is 0.5. The simulation also showed that with this value of K,recovery from a disturbance is very fast. If a disturbance occurs at thestart of a trigger period, then the integral of v_(o) -v_(r) reaches 96%of its steady state value by the end of the period. It is hoped toderive the optimum value of K mathematically at a later time in order tofind out whether it is waveform dependent.

In order to determine a practical algorithm to implement this modulationmethod, and to have automatic compensation for discontinuous current,equation (6) must be expanded into a similar form to equation (4), withno integrations starting from t_(f). There are many expansions thatfulfill these criteria. One that is particularly suitable to themicroprocessor used in example 1 is: ##EQU7##

The time t₀ is a time chosen so that it always corresponds to aparticular phase angle of v_(t). In example 1 it is also chosen to bebefore the start of the trigger period, but this is not essential. Thisis introduced to enable calculations involving v_(t) to be done usingfast look-up tables. In this algorithm, the output voltage is notrequired, but only its integral. This eases transducer requirements, asthe integral can be obtained in digital form directly via an integratingtype v/f converter, an isolating pulse transformer and a counter. It isnot recommended that the integration of the output voltage be done insoftware, as the accumulation of round-off errors could result insubharmonics occurring on the output. In operation, the terms involvingt_(f) are calculated repeatedly from the start of the trigger periodwith t_(f) replaced by the current time, and then added to the otherpre-calculated terms. When the total goes through zero, the thyristor istriggered.

Section (a) of the equation 7 can be calculated any time up to, andperhaps just after, the start of the trigger period. In the prototype itis calculated just after the start to minimise the system response time.Section (b) is found by reading the integral at the start of the triggerperiod then doing the multiplications, and so cannot be done until afterthe start of the trigger period. The remaining sections contain thevariable t_(f), and so must be calculated repeatedly as described above.Section (c) is calculated numerically by successively reading the valueof the integral of v_(o) and adding it to an accumulator.

1.4 Control of Flux and Voltage Boost

In a real motor, the motor flux is not the integral of the appliedvoltage as assumed so far, but is the integral of the applied voltageless the voltage drop across the motor leakage reactance and the statorresistance. If we split the reference voltage, v_(r), into a boostvoltage component, v_(b), to compensate for this voltage drop, and acomponent due to a new reference ψ(t), representing the motor flux attime t, then the integral of v_(r) can be expanded to: ##EQU8##

Using this expansion, equation (7) now becomes: ##EQU9##

For normal motor control without field weakening, the flux waveformshould be kept constant, so ψ(t) in equation (9) can be found from alook-up table. The amplitude and phase of the boost voltage, v_(b), canbe fixed for simple control schemes, or can be varied rapidly as thefundamental component of the motor current changes in response tochanges in load.

1.5 Effect on Ripple Current

The current ripple in each phase of the motor consists of high frequencycomponents only. Because of this, only the leakage inductance of themotor need be considered when determining its value. A per phase motorequivalent circuit that is adequate for determining the current ripplewaveform is shown in FIG. 9(a). Voltage v_(r) is the reference voltageof the corresponding phase of the cycloconverter and is equal to thefundamental component of the back e.m.f. plus the drop across the statorresistance and the total leakage inductance. Using this equivalentcircuit, the ripple current is given by: ##EQU10## A typical ripplecurrent waveform for a positive current from the cycloconverter is shownin FIG. 9(b).

Comparing expression (10) to equations (5) and (6), it can be seen thatthe double pre-integration control method keeps the integral of thecurrent ripple waveform as close as possible to zero during eachtriggering period. This should also keep the amplitude of the currentripple near its minimum value. This indicates that double integrationcontrol is also a very good modulation method for cycloconverters usedin other applications, such as high frequency power system interties andsynchronous motor drives.

1.6 A Simplified Implementation of Double Integration Control

The method of implementing double integration control described above isvery accurate, but requires considerable computing power to implement.If the application is not very demanding, a simpler but much lessaccurate method, which will now be described, can be used.

With some manipulation, equation (5), the basic equation used todetermine the trigger time in double integration control, can beexpressed in the following form: ##EQU11##

In a stable system, i.e. no instability of the type illustrated in FIG.8, the integral of v_(o) -v_(r) at the start of a trigger period is thesame as that at the end of the period (not quite true when V_(r) ischanging, but close enough for this simple implementation), i.e.##EQU12##

Using this relation in equation (11), we obtain the equation: ##EQU13##

The right hand expression in this equation is really a measure of theripple in the integral of v_(o) over the period from t₁ to t₂. It ispositive when the positive thyristor bank is operating and is negativewhen the negative thyristor bank is operating with a magnitude dependingapproximately on the average value of v_(r) during the trigger periodproviding there is no discontinuous current.

This suggests a very simple implementation of double integration controlin which the switching time of the thyristors in the trigger period ischosen to satisfy the following equation: ##EQU14## where M is aconstant which is positive when the positive thyristor bank is operatingand negative when the negative bank is operating and has a magnitudethat is fixed to the expected average magnitude of the expression on theright hand side of equation (13). This is a very rough, but very simpleimplementation. Alternatively, M is varied according to the averagevalue of v_(r) during the period to obtain a more accurateapproximation. It is also possible, although not straight forward, toadjust the magnitude of M to compensate for discontinuous current duringthe period. A very simple version of this method is to set M to zero.

As done in pre-integration control and the first describedimplementation of double integration control, a practical form ofequation (14) can be found by using the voltage v_(t) and ensuring nointegrations start at time t_(f). A form of equation (14) expanded inthis way is: ##EQU15##

The simplest version of the above method with M set to zero actuallyturns out to be similar to pre-integration control, but with animprovement called here "error". This improvement is described below:

From FIG. 6, it can be seen that correct operation of thepre-integration control method relies on firstly that v_(t) remainsundistorted during the trigger period and secondly that the commutationtime of the thyristors is very short. Neither of these conditions maynecessarily hold in a practical cycloconverter. This may result in theunwanted build-up of the integral of v_(o) -v_(r) over several triggerperiods.

This error can be corrected by the following addition to the controlmethod: The voltage v_(o) -v_(r) is fed to an integrator. The output ofthe integrator at time t₂ represents the error in the area between v_(o)and v_(r) at this time. If this error is added to the nextpre-calculated area A in FIG. 5, it will automatically be corrected forin the next trigger period.

Incorporation of this error correction method has the added advantage ofrelaxing the accuracy required in the calculation of the area A. Errorsintroduced by inaccurate calculation will be corrected in the nexttrigger period.

With the error correction method incorporated, equations (2) and (4)become: ##EQU16##

The above method is probably the most degenerate form of doublepre-integration control. Note that it can never degenerate to integralcontrol because continuous integration from time zero (usually thestart-up time of the cycloconverter) of v_(o) and v_(r) is required,whereas in pre-integration control, these integrations are done from thestart of each period.

1.7 Comparison of the Different Implementations of Double IntegrationControl

Below is a description of the different methods of double integrationcontrol in terms of how each method approximates the average of thedifference between the integral of the output voltage and the integralof the reference voltage to zero and how each method preventsinstability. The differences between the methods are illustrated in FIG.20.

The first section of FIG. 20 shows typical output waveforms of the twointegrals when using the method of equation 6. When there is noinstability, this method keeps the averages of the two waveforms exactlyequal (neglecting errors due to the hardware implementation). When thedifference between the integral of V_(o) -V_(r) at times t₁ and t₂ isnot expected to be zero during the current period, indicatinginstability, this difference is reduced at the expense of letting theaverage of the two waveforms change from being equal. The compromisebetween these two requirements is determined by the weighting factor Kin equation 6.

The second section of FIG. 20 shows typical output waveforms of the twointegrals when using the method of equation 14. The average of thedifference between these waveforms is controlled indirectly bycontrolling M. The value of M chosen at the start and end of each periodis usually kept the same, forcing the difference between the integralsof V_(o) -V_(r) at the start and end of the period (which are the valuesof M at the start and end of the period) to zero and thus preventinginstability. When the value of M is changed, for instance when bankcrossover occurs, the difference between the integrals of V_(o) -V_(r)at the start and end of the period is not zero to allow the average ofthe two waveforms to come closer to zero in these situations.

The last section of FIG. 20 shows typical output waveforms of the twointegrals when M is set to zero. How closely the average of thedifference between the two waveforms approximates zero now depends onthe amount of ripple in the integral of V_(o). Stability is forced byalways setting the difference between the integrals of V_(o) -V_(r) atthe start and end of each period to zero.

2. IMPROVEMENT TO VOLTAGE RANGE AND DISTORTION BEHAVIOUR

Using the modulation method of equation 6 allows the 3 pulsecycloconverter to efficiently and accurately control the speed of aninduction motor. In the cycloconverter of Example 1 some furthermodifications were made to maximise its performance and these will bedescribed below. With these modifications, the maximum output voltagebefore clipping is increased to 95% of the input voltage and thedistortion when operating at or near maximum output voltage is improved.The modifications can be used with most modulation methods, includingthe improved and prior art modulation methods described herein.

2.1. Improvement of Output Voltage Range by Changing Output NeutralVoltage

The basic circuit of a cycloconverter with a 3 phase induction motorload (assumed here to be star connected) is shown in FIG. 11. Normallythe neutral voltage, v_(n), is kept as close to zero as possible, but inactual fact, v_(n) can be any value without affecting the motor,provided the voltages between U, V and W are 3 phase sine waves. Bychoosing a suitable waveform for v_(n), it is possible to reduce thepeak voltage on the output of the cycloconverter for the same line toline voltage. The normal waveform chosen for v_(n) is a sine wave offrequency 3 times the output frequency and an amplitude that willminimise the peak voltage on the outputs. This procedure is well knownand has been documented many times, an example being Nakajima et al["Reactive Power Reduced Cycloconverter with Bias Voltage at the NeutralPoint"--IEEE--IAS Meeting--1980--Pt 2, pp. 785-790].

The same method to improve the output voltage range is used here, butinstead of a sine wave, the waveform is chosen to maximise the effect.FIG. 12(a) shows the waveform used for v_(n) and how it is chosen andFIG. 12(b) shows the resulting output waveforms. With this modification,the peak line to line output voltage before clipping is 95% of the inputvoltage. With the normal method of choosing a sine wave for v_(n), thepeak line to line output voltage is also improved to 95% of the inputvoltage, but the output instantaneous voltage is at its peak level for alonger proportion of each cycle resulting in any clipping producing moresevere output voltage distortion.

2.2 Improving Distortion by Changing Input Reference

So far it has been assumed that the measuring reference point used bythe cycloconverter control circuits is the input neutral point. As thereis no actual neutral supplied to the cycloconverter, this referencepoint would have to be obtained using a star network of resistors. Inthe prototype cycloconverter, an alternative reference point, obtainedwith the circuit of FIG. 13, is used. The voltage waveform at this newreference point with respect to the true input neutral point is the sameas v_(n) in FIG. 12(a) with a frequency of three times the mainsfrequency. With respect to this new reference point, the input waveformsare no longer sine waves, but are the same as the waveforms of FIG.12(b).

The advantage of using the alternate reference is that when operating atmaximum output voltage, the voltage distortion is reduced and the ripplefrequency is doubled. This effect is shown in FIG. 14. An important sidebenefit to using the alternate reference is that the input waveforms canbe approximated by trapezoidal waveforms. This considerably eases thecalculations required in the microprocessor to determine the start andend times of a trigger period.

3. IMPLEMENTATION--EXAMPLE 1 3.1 Motor Requirements

In developing the modulation method of the present invention, the motorrequirements when driven by a cycloconverter were looked at verycarefully, particularly for the induction motor, as this is thepreferred motor for most applications.

One way to get good performance out of an inductionmotor--cycloconverter drive is to make it simulate as close as possiblea thyristor converter--D.C. motor drive, and this is the approach takenhere. In a D.C. drive, the motor flux is kept constant by a constantfield current, while the speed is controlled by a highly distorted D.C.voltage applied to the armature. The distortion in the armature voltageproduces a high ripple current and a corresponding increase in motorheating, but has little effect on the motor performance. This is becausethe torque is proportional to the product of flux and current and so theripple current produces only a corresponding high frequency rippletorque without affecting the average torque.

To simulate the conditions in a D.C. machine for an induction motor, thecomponents of flux linkage in the three phases must be kept as close aspossible to three sine waves of equal amplitude and displaced by 120degrees, which means that the integrals of the three input voltages mustbe likewise kept. This is the criteria on which the new modulationmethod was developed. If the flux linkage components in each phase canbe kept sinusoidal by the cycloconverter, then only the component ofcurrent of the same frequency can contribute to the D.C. component oftorque. There would be a pulsating torque due to the ripple componentsof the currents, but this would be no worse than that of the equivalentD.C. machine containing 2/3 the number of thyristors as thecycloconverter.

3.2 Hardware

FIG. 15 is a block diagram of Example 1 using a three pulsecycloconverter with double integration control. The zero currentdetectors on each output phase work by sensing the voltage across eachthyristor as described by Hamblin and Barton ["Cycloconverter ControlCircuits"--IEEE Trans. Ind. App. 1972 Vol. IA-8--No. 4, p. 443-452]. Tomeasure the integral of the output voltages, three voltage to frequencyconverters of the integrating type interfaced to the microprocessor viacounters are used. An offset voltage (not shown in FIG. 15) is appliedto the input of the voltage to frequency converters to enable them tooperate in the bipolar mode. This is compensated for by themicroprocessor software. The input analogue speed command is alsomeasured by a voltage to frequency converter coupled to a counter. Thishas the advantage of being cheaper than an analogue to digitalconverter, allows the average speed over each speed sampling period tobe measured rather than the speed at each sampling instant, and givesinfinite speed resolution.

The microprocessor is timed by two interrupt signals supplied by a phaselocked loop locked to the mains. One is at the same frequency as themains and is used to synchronise the microprocessor to the mains. Theother is at 60 times the mains frequency and determines the samplinginstances.

For accurate control of voltage boost to enable accurate and fast motorresponse, the tachometer can be added as shown. In Example 1 no currentfeedback is used as the cycloconverter thyristors are fuse protected,and the motor current can be deduced from the slip (derived from thetacho feedback).

3.3 Microprocessor Requirements

The main limitation on the choice of microprocessor is processing speed.As can be seen from equation (7), a large number of calculations arerequired during each sampling interval, as well as other jobs such aschecking for zero current. To reduce the load on the microprocessor, thesampling interval should be as long as possible, but a longer samplinginterval produces extra voltage distortion because the thyristor firingtime can be delayed by up to one sampling interval from the ideal time.It would be ideal if the sampling interval of v_(o) that is introducedby delaying the firing time by one sampling interval is much less thanthe normal ouput distortion, which can be quantified as the normal peakvalue of the integral of v_(o) -v_(r). The sampling interval chosen forthe prototype is 333 microseconds which results in an error of about onequarter of the normal output distortion. This is greater than the ideal,but was limited by the speed of the microprocessor used.

The microprocessor chosen, the TMS32010, is one of the very few on thepresent market with enough processing speed without resorting to bitslice devices. It is designed for digital signal processing, but has aninstruction set powerful enough for general control use.

3.4 Performance of Example 1

Drives using cycloconverters are known for their smoothness at lowspeeds. With the improved method of modulation, this is improved evenfurther. The double integration control technique virtually, if notentirely, eliminates any possibility of subharmonics and preventsdistortion being introduced by discontinuous currents.

The output frequency of Example 1 is capable of going up to at least 25Hz and the output voltage is 95% of the input voltage. With a two poleinduction motor, this allows a speed range from 0 to 1500 r.p.m. whichis adequate for most applications. Note that a maximum output frequencyof 50 Hz can be obtained from a 6-pulse cycloconverter using doublepre-integration control, but at the expense of twice the number ofthyristors in the power circuit. A standard mains voltage deltaconnected induction motor can be used for the 3-pulse cycloconverter byreconnecting it to star configuration. The line to line voltage requiredfor 25 Hz operation would then be 86.6% of the mains voltage, which is areasonable match to the cycloconverter. This is what was used fortesting Example 1.

The motor used for testing was a 4 pole, 7.5 kW motor which was loadedto 2 kW at 25 Hz by a DC generator. No tachometer feedback was used andthe voltage boost from the cycloconverter was fixed at a level thatwould give a maximum torque at low speed of one half full load torque.From 0.5 to 25 Hz, the highest frequency tested, the drive performancewas excellent with no hint of instability or torque pulsations. Below0.5 Hz, multiple switchings between the positive and negative banksoccurred near each true current zero point, producing slight torquepulsations at these instances. The reason for this has yet to beinvestigated.

The response time of the cycloconverter with double pre-integrationcontrol depends on how the algorithms are implemented in themicroprocessor. In a preferred embodiment, as explained in section 1.3,the first part of equation (7) for a given trigger period is calculatedjust after the start of that period. To do this calculation, thereference voltage waveform to the end of the period must be known. Thisproblem is overcome in Example 1 by measuring the input variables, speedreference and tachometer output, or just the speed reference when thereis no tacho feedback, every 120 degrees advance of input phase, butdelaying the use of these readings until after the next 120 degreesadvance. This gives an effective response time delay of about 7milliseconds, which is as good as the best D.C. drives.

4. BANK CROSS OVER DETERMINATION 4.1 Bank Switching--First Method

A simple method of determining the time when bank cross-over shouldoccur that can be used when the modulation method compensates fordiscontinuous current (e.g. pre-integration control and doublepre-integration control) is as follows:

If, for example, in FIG. 6, the next thyristor is not switched on by theend of the trigger period at t₂, then it is not possible to maintain theaverage output at the reference voltage and this is the time when bankswitching to the negative bank is carried out. To calculate the nexttriggering time after the bank cross-over, the time of bank cross-overcan be made the starting time of but need not be the next triggerperiod. Note that this is a very different approach to that used inpresent modulation schemes which switch banks at an estimate of the zerocrossibg of the fundamental component of output current. Instead theinstant of bank switching is selected to minimise the output voltagedistortion.

4.2 Bank Switching--Improved Method

The optimum bank cross-over time for the majority of Induction Motorapplications is the first time the actual current is zero (and thus allthyristors in that phase are off) after the fundamental component ofcurrent passes through zero. Since the instantaneous value of thecurrent ripple is proportional to the integral of v_(o) -v_(r), which isa value which is available when the double pre-integration controlmethod is used, it is quite easy to determine accurately this optimumbank cross-over time. As shown in FIG. 10, for a positive outputcurrent, the cross-over to the negative bank should occur at the firstinstant when the output current is zero and the integral of v_(o) -v_(r)is positive (the integral should be negative for a negative outputcurrent). This is the first point when the current is zero and thefundamental component of the current is negative. To calculate the nexttriggering time after the bank cross-over, the time of bank cross-overfor pre-integration and double pre-integration control can be made thestarting time of but need not be the next trigger period.

This bank cross-over determination method is an improvement over themethod described in section 4.1. Bank cross-over is initiated withoutwaiting for the end of the trigger period in which the true current zerooccurs.

When the cycloconverter has a 3 phase load with no neutral connection,as in the case with an induction motor load, an extra modification canbe made to take account of the fact that any voltage distortion that iscommon to all three outputs does not produce any corresponding currentripple. The instantaneous value of the current ripple in a particularoutput is in this case proportional to the integral of v_(o) -v_(r) lessthe instantaneous average value of the corresponding integrals on eachof the three outputs. This is expressed in equation (17): ##EQU17## Tocompensate for this, the cross-over from the positive to the negativebank should now occur at the first instant when the output current iszero and the expression on the right hand side of equation (17) ispositive (and negative for a negative to positive bank cross-over). Thisis the scheme used in Example 1.

Note that this bank switching method can be used for pre-integrationcontrol and double integration control as well as for most other controlmethods (including the prior art methods disclosed herein).

5. POWER FACTOR IMPROVEMENT

Cycloconverters are well known for their poor input power factor,particularly at low output voltages. The power factor improvement methoddescribed here is well suited for use with double integration control,but can also be used with other modulation methods. The trade-off isthat subharmonic components of the input current may appear. Thefollowing description of the method assumes the cycloconverter is a 3pulse, 18 tyristor type (as shown in FIG. 2 for example).

5.1 Description

A basic circuit of a cycloconverter with a 3 phase induction motor load(assumed here to be star connected) is shown in FIG. 11. As explainedpreviously, the neutral voltage, v_(n), can be any value withoutaffecting the motor, provided the voltages between U, V and W are 3phase sine waves. The new power factor improvement method simply choosesv_(n) to maximise the input power factor. This is similar to thetechnique used by Nakajima et al [previously referenced], but is muchmore effective.

To maximise the input power factor, the neutral voltage is changed asfollows. The three phase voltages of a three phase sinusoidal waveformreferenced to the neutral are shown in FIG. 17(a). If the reference ischanged to the most negative voltage of FIG. 17(a) at any instant (whichof course will not affect the line to line voltages in any way) thewaveforms in FIG. 17(b) are obtained. This output reference can now bemade equal to the most negative instantaneous input voltage, as shown inFIG. 18(a). The neutral voltage, v_(n), will now be a combination of thetwo new reference waveforms in FIGS. 17(a) and 18(a). To make visualinterpretation easier, this input reference is changed to a "straightline" reference in FIG. 18(b). Also in FIG. 18(b), the referencevoltage, v_(r), of one phase of a possible output waveform issuperimposed on the input waveform in order to show the relation betweenthe two.

5.2 Expected Improvements

To show the effect of change in v_(n) on the input power factor, thethree output waveforms and the input current waveform of one phase areshown in FIG. 19 for a very low output voltage, which is normally whenthe input power factor is the worst. Note that for simplicity, thethyristor commutation time is assumed to be zero. As can be seen in FIG.19(d), the input current is zero for most of the time and is only equalto one or the addition of two of the output currents during relativelyshort intervals. When the input current is zero, the output currents areactually "free-wheeling" through the thyristors, rather than circulatingaround the input phases. As the amplitude of the output referencevoltage reduces to zero, the intervals during which the input current isnot zero, reducing also the r.m.s. value of the input current to zero.This can be compared to the normal method of keeping v_(n) to a minimum,where there is no "free-wheeling" current and the r.m.s. values of theinput and output currents are always approximately the same. Since theinput power is always equal to the output power, the input power factoris improved by the same factor by which the r.m.s. input current hasbeen reduced.

An improvement is also made in the output voltage distortion. When theoutput voltage is low, the sections of the input voltage waveforms thatare applied to the output tend to be the low amplitude sections. Withthe normal method, the output is composed of sections from the peaks ofthe input voltage waveforms, resulting in a higher r.m.s. voltageripple.

The maximum output voltage with the new scheme is shown in FIG. 17(b).The maximum peak value of v_(r) is 3/π or about 95% of the peak value ofthe input voltage, which is the same as can be obtained without powerfactor improvement.

5.3 Input Subharmonics

The problem of input subharmonics can be seen clearly in FIG. 18(d).Here, a D.C. component of the input current is present. In this example,the output frequency is one half the input frequency, so the D.C.component is probably due to the f_(i) -2f_(o) intermodulation product.

The presence of input subharmonic currents is likely to be a problemonly if the mains transformer feeding the cycloconverter is near therating of the cycloconverter. In this situation, the transformer may gotoo far into saturation.

APPENDIX I Determination of Stability Constant, K

To get a rough idea of the optimum value of the stability constant, K,an approximate stimulation of one output of the cycloconverter wasundertaken. The input and output waveforms were approximated to thewaveforms shown in FIG. 19. Time and voltage are assumed to benormalised to the values shown in this figure. It is assumed thatpositive current is flowing from the output and that the outputreference voltage is constant at 0.5.

For the trigger period t₁ to t₂ as shown in FIG. 16, equation (6)becomes:

    0=t.sub.α.sup.2 -t.sub.α (1.5+2K)+0.5K+(1/6)+2A (A1)

where ##EQU18##

Solving this for t.sub.α : ##EQU19##

The value of A for the start of the next period is given by: ##EQU20##

The variation in the value of A from period to period is a goodindication of stability. To determine stability, A was initially set tozero and then calculated for each subsequent period for different valuesof K. The results are listed below:

                  TABLE A1                                                        ______________________________________                                        Simulation Results                                                            Period elapsed                                                                             Value of A (% of final value)                                    ______________________________________                                        K = 0.4                                                                       0            0                                                                1            106.515142953                                                    2            99.254741406                                                     3            100.0825289                                                      4            99.990826713                                                     5            100.001019211                                                    6            99.999886754                                                     K = 0.5                                                                       0            0                                                                1            96.598008990                                                     2            99.995790914                                                     3            99.999999993                                                     4            100.000000000                                                    5            100.000000000                                                    6            100.000000000                                                    K = 0.6                                                                       0            0                                                                1            88.323517689                                                     2            98.9014162177                                                    3            99.899798382                                                     4            99.990888016                                                     5            99.999171616                                                     6            99.999924695                                                     ______________________________________                                    

The above results indicate that the best value of K is about 0.5. Thisgives the fastest settling time to equilibrium conditions.

I claim:
 1. In a static power frequency changer connecting one or moreinput phases to one or more outputs,said changer comprising one or moreelectronic switching means comprising a plurality of electronicswitches, modulation means to sequentially activate individual switchesof said electronic switching means, said electronic switching meansconnecting an AC voltage supply comprising said one or more input phasesto an output of said one or more outputs, so that the output voltagewaveform (v₀) at said output is built up of sections of input voltagewaveforms on said one or more input phases; a method of selecting, foreach said output, an instant of switching (t_(f)) of the input waveformto be connected to said output, wherein: for each output said instant ofswitching is chosen so that the average over a predetermined timeinterval of the difference between the continuous integral of thedesired output voltage (v_(r)) and an estimate of the continuousintegral of the actual output voltage is minimised, said predeterminedtime interval including said instant of switching to another inputwaveform.
 2. The method of claim 1, said method excluding the specialcase of equations (14) or (15) having M=0 for each trigger period,equation (14) being: ##EQU21## where M is a constant, and equation (15)being: ##EQU22## where M is a constant,v_(t) is the input voltageconnected to the electronic switch to be triggered in a trigger period˜, t₁ is the start of said predetermined time interval, and t₂ is theend of said predetermined time interval.
 3. The method of claim 2wherein said predetermined time interval does not necessarily end atsaid instant of switching.
 4. The method of claim 3 wherein saidpredetermined time interval includes only one said instant of switchingdetermined by said method.
 5. The method of claim 4 wherein saidpredetermined time interval is a representative one of a plurality ofidentical consecutive predetermined time intervals, and the end of eachpredetermined time interval coincides with the start of a nextpredetermined time interval.
 6. The method of claim 5 wherein the startof said predetermined time interval is defined as the time ofintersection of a first input voltage waveform with said desired outputvoltage waveform and the end of said predetermined time interval isdefined as the time of intersection of a second input voltage waveformwith said desired output voltage waveform;said first input voltagewaveform being the last input voltage waveform connected to said outputprior to said instant of switching; and said second input voltagewaveform being the input voltage waveform to be connected to said outputat said instant of switching.
 7. The method of claim 6, said methodfurther including the provision of system stabilising means to renderthe output waveform stable.
 8. The method of claim 7 wherein said systemstabilising means comprises means to minimise or eliminate oscillationof the continuous integral of the difference between said output voltagewaveform and said desired output voltage waveform at the end ofconsecutive ones of said predetermined time interval.
 9. The method ofclaim 8 wherein the instant of switching is chosen so that the integralfrom the start (t₁) to the end (t₂) of said predetermined time intervalof the difference between said output voltage waveform and said desiredoutput voltage waveform is minimized.
 10. The method of claim 9 whereinsaid average is minimised and said integral of said difference isminimised by finding a solution to equation (6), equation (6) being##EQU23## where K is a constant, the choice of which depends on thedegree of stability required.
 11. The method of claim 10 wherein theequation of claim 10 is expanded so that no integrations start from saidtime of switching.
 12. The method of claim 11 wherein the equation ofclaim 10 is expanded to equation (7), equation (7) being: ##EQU24## 13.The method of claim 9 wherein said average is minimised and saidintegral of said difference is minimised by finding a solution toequation (13), equation (13) being: ##EQU25##
 14. The method of claim 13wherein the equation of claim 13 is expanded so that no integrationsstart from said time of switching.
 15. The method of claim 14 wherein anapproximation is made to the RHS of equation (13).
 16. The method ofclaim 13 wherein an approximation is made to the RHS of equation (13).17. The method of claim 3, said method further including the provisionof system stabilising means to render the output waveform stable. 18.The method of claim 17 wherein said system stabilising means comprisesmeans to minimise or eliminate oscillation of the continuous integral ofthe difference between said output voltage waveform and said desiredoutput voltage waveform at consecutive ends of ones of saidpredetermined time interval.
 19. The method of claim 18 wherein theinstant of switching is chosen so that the difference between the valuesof the continuous integral of the difference between said output voltagewaveform and said desired output voltage waveform at the end of saidpredetermined time interval and at a previous end of a selected one ofsaid ones of said predetermined time interval is minimised; andwhereinthe degree of minimisation of said average with respect to the degree ofminimisation of said difference is chosen according to the stabilityrequirements of the specific application.
 20. The method of claim 19wherein said predetermined time interval includes only one said instantof switching.
 21. The method of claim 2 wherein said plurality ofelectronic switches comprises naturally commutated thyristors.
 22. Themethod of claim 21 wherein said predetermined time interval does notnecessarily end at said instant of switching.
 23. The method of claim 22wherein said predetermined time interval includes only one said instantof switching determined by said method.
 24. The method of claim 23wherein said predetermined time interval is a representative one of aplurality of identical consecutive predetermined time intervals, and theend of each predetermined time interval coincides with the start of anext predetermined time interval.
 25. The method of claim 24 wherein thestart of said predetermined time interval is defined as the time ofintersection of a first input voltage waveform with said desired outputvoltage waveform and the end of said predetermined time interval isdefined as the time of intersection of a second input voltage waveformwith said desired output voltage waveform;said first input voltagewaveform being the last input voltage waveform connected to said outputprior to said instant of switching; and said second input voltagewaveform being the input voltage waveform to be connected to said outputat said instant of switching.
 26. The method of claim 25, said methodfurther including the provision of system stabilising means to renderthe output waveform stable.
 27. The method of claim 26 wherein saidsystem stabilising means comprises means to minimise or eliminateoscillation of the continuous integral of the difference between saidoutput voltage waveform and said desired output voltage waveform at theend of consecutive ones of said predetermined time interval.
 28. Themethod of claim 27 wherein the instant of switching is chosen so thatthe integral from the start (t₁) to the end (t₂) of said predeterminedtime interval of the difference between said output voltage waveform andsaid desired output voltage waveform is minimized.
 29. The method ofclaim 28 wherein said average is minimised and said integral of saiddifference is minimised by finding a solution to equation (6), equation(6) being: ##EQU26## where K is a constant, the choice of which dependson the degree of stability required.
 30. The method of claim 29 whereinthe equation of claim 29 is expanded so that no integration start fromsaid time of switching.
 31. The method of claim 30 wherei the equationof claim 29 is expanded to equation (7), equation (7) being: ##EQU27##32. The method of claim 28 wherein said average is minimised and saidintegral of said difference is minimised by finding a solution toequation (13), equation (13) being: ##EQU28##
 33. The method of claim 32wherein the equation of claim 32 is expanded so that no integrationsstart from said time of switching.
 34. The method of claim 33 wherein anapproximation is made to the RHS of equation (13).
 35. The method ofclaim 32 wherein an approximation is made to the RHS of equation (13).36. The method of claim 22, said method further including the provisionof system stabilising means to render the output waveform stable. 37.The method of claim 36 wherein said system stabilising means comprisesmeans to minimise or eliminate oscillation of the continuous integral ofthe difference between said output voltage waveform and said desiredoutput voltage waveform at consecutive ends of ones of saidpredetermined time interval.
 38. The method of claim 37 wherein theinstant of switching is chosen so that the difference between the valuesof the continuous integral of the difference between said output voltagewaveform and said desired output voltage waveform at the end of saidpredetermined time interval and at a previous end of a selected one ofsaid ones of said predetermined time interval is minimised; andwhereinthe degree of minimisation of said average with respect to the degree ofminimisation of said difference is chosen according to the stabilityrequirements of the specific application.
 39. The method of claim 38wherein said predetermined time interval includes only one said instantof switching.
 40. A static power frequency changer connecting one ormore input phases to one or more outputs,said changer comprising one ormore electronic switching means comprising a plurality of electronicswitches, modulation means to sequentially activate individual switchesof said electronic switching means, said electronic switching meansconnecting an AC voltage supply comprising said one or more input phasesto an output of said one more outputs, so that the output voltagewaveform (v₀) at said output is built up of sections of input voltagewaveforms on said one or more input phases; said modulation meansincluding means for selecting, for each said output, an instant ofswitching (t_(f)) of the input waveform to be connected to said output,wherein: for each output said instant of switching is chosen so that theaverage over a predetermined time interval of the difference between thecontinuous integral of the desired output voltage (v_(r)) and anestimate of the continuous integral of the actual output voltage isminimised, said predetermined time interval including said instant ofswitching to another input waveform.
 41. The changer of claim 40,wherein said modulation means excludes the special case of equations(14) or (15) having M=0 for each trigger period, equation (14) being:##EQU29## where M is a constant, and equation (15) being: ##EQU30##where M is a constant,v_(t) is the input voltage connected to theelectronic switch to be triggered in a trigger period t₁ is the start ofsaid predetermined time interval, and t₂ is the end of saidpredetermined time interval.
 42. The changer of claim 41 wherein saidpredetermined time interval does not necessarily end at said instant ofswitching.
 43. The changer of claim 42 wherein said predetermined timeinterval includes only one said instant of switching determined by saidmethod.
 44. The changer of claim 43 wherein said predetermined timeinterval is a representative one of a plurality of identical consecutivepredetermined time intervals, and the end of each predetermined timeinterval coincides with the start of a next predetermined time interval.45. The changer of claim 44 herein the start of said predetermined timeinterval is defined as the time of intersection of a first input voltagewaveform with said desired output voltage waveform and the end of saidpredetermined time interval is defined as the time of intersection of asecond input voltage waveform with said desired output voltagewaveform;said first input voltage waveform being the last input voltagewaveform connected to said output prior to said instant of switching;and said second input voltage waveform being the input voltage waveformto be connected to said output at saad instant of switching.
 46. Thechanger of claim 45, said changer further including the provision ofsystem stabilising means to render the output waveform stable.
 47. Thechanger of claim 46 wherein said system stabilising means comprisesmeans to minimise or eliminate oscillation of the continuous integral ofthe difference between said output voltage waveform and said desiredoutput voltage waveform at the end of consecutive ones of saidpredetermined time interval.
 48. The changer of claim 47 wherein theinstant of switching is chosen so that the integral from the start (t₁)to the end (t₂) of said predetermined time interval of the differencebetween said output voltage waveform and said desired output voltagewaveform is minimized.
 49. The changer of claim 48 wherein said averageis minimised and said difference is minimised by finding a solution toequation (6), equation (6) being: ##EQU31##
 50. The changer of claim 49wherein the equatin of claim 49 is expanded so that no integrationsstart from said time of switching.
 51. The changer of claim 50 whereinthe equation of claim 49 is expanded to equation (7), equation (7)being: ##EQU32##
 52. The changer of claim 48 wherein said average isminimised and said integral of said difference is minimised by finding asolution to equation (13), equation (13) being: ##EQU33##
 53. Thechanger of claim 52 wherein the equation of claim 52 is expanded so thatno integraations start from said time of switching.
 54. The changer ofclaim 53 wherein an approximation is made to the RHS of equation (13).55. The changer of claim 52 wherein an approximation is made to the RHSof equation (13).
 56. The changer of claim 42, said changer furtherincluding the provision of system stabilising means to render the outputwaveform stable.
 57. The changer of claim 56 wherein said systemstabilising means comprises means to minimise or eliminate oscillationof the continuous integral of the difference between said output voltagewaveform and said desired output voltage waveform at consecutive ends ofones of said predetermined time interval.
 58. The changer of claim 57wherein the instant of switching is chosen so that the differencebetween the values of the continuous integral of the difference betweensaid output voltage waveform and said desired output voltage waveform atthe end of said predetermined time interval and at a previous end of aselected one of said ones of said predetermined time interval isminimised; andwherein the degree of minimisation of said average withrespect to the degree of minimisation of said difference is chosenaccording to the stability requirements of the specific application. 59.The changer of claim 58 wherein said predetermined time intervalincludes only one said instant of switching.
 60. The changer of claim 41wherein said plurality of electric switches comprises naturallycommutated thyristors.
 61. The changer of claim 60 wherein saidpredetermined time interval does not necessarily end at said instant ofswitching.
 62. The changer of claim 61 wherein said predetermined timeinterval includes only one said instant of switching determined by saidmethod.
 63. The changer of claim 62 wherein said predetermined timeinterval is a representative one of a plurallity of identicalconsecutive predetermined time intervals, and the end of eachpredetermined time interval coincides with the start of a nextpredetermined time interval.
 64. The changer of claim 63 herein thestart of said predetermined time interval is defined as the time ofintersection of a first input voltage waveform with said desired outputvoltage waveform and the end of said predetermined time interval isdefined as the time of intersection of a second input voltage waveformwith said desired output voltage waveform;said first input voltagewaveform being the last input voltage waveform connected to said outputprior to said instant of switching; and said second input voltagewaveform being the input voltage waveform to be connected to said outputat said instant of switching.
 65. The changer of claim 64, said changerfurther including the provision of system stabilising means to renderthe output waveform stable.
 66. The changer of claim 65 wherein saidsystem stabilising means comprises means to minimise or eliminateoscillation of the continuous integral of the difference between saidoutput voltage waveform and said desired output voltage waveform at theend of consecutive ones of said predetermined time interval.
 67. Thechanger of claim 66 wherein the instant of switching is chosen so thatthe integral from the start (t₁) to the end (t₂) of said predeterminedtime interval of the difference between said output voltage waveform andsaid desired output voltage waveform is minimized.
 68. The changer ofclaim 67 wherein said average is minimised and said difference isminimised by finding a solution to equation (6), equation (6) being:##EQU34## where K is a constant, the choice of which depends on thedegree of stability required.
 69. The changer of claim 68 wherein theequation of claim 68 is expanded so that no integrations start from saidtime of switching.
 70. The changer of claim 69 wherein the equation ofclaim 68 is expanded to equation (7), equation (7) being: ##EQU35## 71.The changer of claim 67 wherein said average is minimised and saidintegral of said difference is minimised by finding a solution toequation (13), equation (13) being: ##EQU36##
 72. The changer of claim71 wherein the equation of claim 71 is expanded so that no integrationsstart from said time of switching.
 73. The changer of claim 72 whereinan approximation is made to the RHS of equation (13).
 74. The changer ofclaim 71 wherein an approximation is made to the RHS of equation (13).75. The changer of claim 61 wherein said static power frequency changeris connected as a three pulse three input phase three output phasecycloconverter.
 76. The changer of claim 61 wherein said static powerfrequency changer is connected as a six pulse three input phase threeoutput phase cycloconverter.
 77. The changer of claim 61 wherein saidstatic power frequency changer is connected as a two pulse single inputphase two output phase cycloconverter.
 78. The changer of claim 61, saidchanger further including the provision of system stabilising means torender the output waveform stable.
 79. The changer of claim 78 whereinsaid system stabilising means comprises means to minimise or eliminateoscillation of the continuous integral of the difference between saidoutput voltage waveform and said desired output voltage waveform atconsecutive ends of ones of said predetermined time interval.
 80. Thechanger of claim 79 wherein the instant of switching is chosen so thatthe difference between the values of the continuous integral of thedifference between said output voltage waveform and said desired outputvoltage waveform at the end of said predetermined time interval and at aprevious end of a selected one of said ones of said predetermined timeinterval is minimised; andwherein the degree of minimisation of saidaverage with respect to the degree of minimisation of said difference ischosen according to the stability requirements of the specificapplication.
 81. The changer of claim 80 wherein said predetermined timeinterval includes only one said instant of switching.